The number pi, or π, cannot be stated using conventional digits.  It can only be approximated.  The approximation begins with 3.14, so today, March 14, is “Pi Day”.  The approximation continues ……159, so 1:59 on March 14 would be “Pi Minute”.  If you wait 26 ticks of the clock after you reach Pi Minute, you would be in “Pi Second” – 3.1415926.

One of the oddities of the number is that you (or rather the nearest mathematician) can keep calculating to “Pi Millisecond” or “Pi Microsecond” or any level of precision you like.  The number never resolves.  The sequence of digits never ends.

A couple of years ago, I read an article about a pair of brothers whose life work was to calculate pi to the greatest precision possible.  They wanted to see if anywhere in this sequence of billions of digits there might be a pattern, some sequence that repeats.  They had found nothing at the time of writing.

According to the article, the brothers had calculated pi to such a level of precision that if you compared two circles, one a perfect circle with a circumference equal to that of the entire universe, the other a circle that you would get using their estimation of pi, the two circles spanning the universe would differ by the width of a molecule.  Yet, the brothers or their descendants could continue to calculate for a century, or a millennium, or for as long as the earth lasts, and never fully state the number.  It cannot be stated using digits.

It is an irrational number, meaning that it cannot be stated as the ratio of two whole numbers.  No great shock here.  It is also a transcendental number, meaning that there is no algebraic equation whose coefficients are rational numbers that will yield pi as the answer.

Yet this irrational, transcendental, incalculable number is present in nature in strange ways.  If you were to make a chart that showed the hours of daylight for each day over a year, the result would be a sine wave, a function dependent on pi.  The planets revolve around the sun along lines that can be calculated using pi.

Most astounding to me is the number’s appearance in one of the most profound and beautiful statements of mathematics, one that combines seven basic symbols in the strangest way.

e^iπ  + 1 = 0

“e” is “Euler’s number”, the “natural” logarithm.  It can be stated as the value of (1 + 1/n)ⁿ as n approaches infinity.  Like pi, it is irrational and transcendental.  Like pi, it can be calculated forever without resolving.

“i” is an “imaginary” number, the square root of (-1).

We take the exotic, irresolvable number pi and multiply it by the square root of the number (-1).  We then raise “e”, Euler’s number, which is also exotic and irresolvable, to the power of the product of i and pi.  That number, somehow, is negative 1.  Add the number 1 to get zero.

It takes but a few paragraphs in the hands of someone who understands these abstractions to explain the basis for this profound identity. I do not understand it, but while I am actively reading an explanation, I can glimpse how it might all make sense to someone who has achieved mathematical satori.  It’s like standing in a desert at night and seeing flashes of lightning from a storm that lies over the horizon.  Something is being illuminated, but it is far in the distance and instantly returns to darkness.

It is a profound statement about the strangeness of the universe we inhabit that these two numbers, neither of which can be stated using the digits familiar to our daily lives, when combined with a number that cannot exist in nature, produce the most prosaic number we can conceive.